How to correct my code for solving the Josephus problem?

Here's my solution using pattern-matching:

Range[10] //. {x_, y_, z___} :> {z, x}

{5}

NestWhile[Rest @ RotateLeft @ # &, Range @ 10, Length @ # > 1 &]

{5}

FixedPoint[If[Length @ # > 1, Rest @ RotateLeft[#], #] &, Range @ 10]

Edit

Historical note: As far as I can remember, Josephus roulette (a plain treason to his companions) consisted of killing every third person.

FixedPoint[If[Length@# != 1, Rest@RotateLeft[#, 2], #] &, Range@10]

{4}

Note: The direction is important. RotateRight[] will select another victim.

You can use Nest and define a function so you don't have to know the number of iterations:

josephus[n_] := Nest[Rest@RotateLeft[#] &, Range@n, n - 1]

So

josephus[10]

{5}

josephus[200]

{145}